plotcoord1nelem coord2nelem1sigma elseif norder2 plotcoord122nelem

Plotcoord1nelem coord2nelem1sigma elseif norder2

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plot([coord(1:nelem) coord(2:nelem+1)]’,sigma’) elseif norder==2, plot([coord(1:2:2*nelem) coord(2:2:2*nelem+1) coord(3:2:2*nelem+2)]’,sigma’) end xlabel(’x’), ylabel(’\sigma’); (b) ii=pos(2,:) ; u = sol(nonzeros(ii)) ; u = [0.2 ; 0.4] (c) ii=dest(3,:) ; u=sol(ii) ; u=0.4
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36 Answers to the exercises of chapter 14 14.8 (a) d dx c du dx - f = 0 u (0) = 5 u ( l ) = 5 (b) u ( x ) = f 2 c x 2 - fl 2 c x + 5 (c) When the thickness t = 5mm the glucose level in the middle of the construct will become less than zero (in reality of course zero) meaning that eventually cells will die in the middle of the construct. 14.9 (a) Add: xx=n(int,:)*nodcoord; a1=1.6; % in [cm] a2=0.15; % in [cm^-1] a3=0.8; % dimensioneless E=10; % in [Ncm^-2] r=a1*(sin(a2*xx+a3))^3; A=pi*r^2 %calculate surface of cross section c=E*A; % product of cross section and Youngs modulus in the element routine. (b) See figure 14.3 .
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Exercises 37 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 x u Fig. 14.3. Exercise 14.9: Displacement as a function of x
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15 Answers to the exercises of chapter 15 Exercises 15.1 (b) See Fig. 15.1 . For small a the solution is approximately the same as the solution of the diffusion equation. For larger a particles drag along with the fluid and thus the higher con- centrations shift to the right. For values higher than 10 the solution becomes unstable. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x u a = 0.001 (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x u a = 1 (b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 x u a = 10 (c) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x u a = 20 (d) Fig. 15.1. Solution of the steady convection diffusion equation for different values of a using 5 linear elements 38
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Exercises 39 (c) With h = 0 . 2 and c = 1 it follows that Pe h > 1 if a > 10. 15.2 (a) It can be seen that using θ = 0 . 5 the solutions remain stable, even for large time steps. When θ < 0 . 5 the solution starts to oscillate when the time step becomes to large. (b) When a > 20 solutions will become unstable. (c) When Δ t = 0 . 001 the discretization of u no longer allows an ”exact” description of the u as a function of x . (d) Reducing the convective velocity does not influence the sta- bility much. Increasing the number of elements (i.e. reducing the size of an element) again leads to stable solutions. 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 x u Fig. 15.2. Solution for different time steps 15.3 (a) Initial conditions. Use sol(:,1) (second index is the time step) For initial conditions the following script can be used: for inode = 1: nnodes idof=dest(inode,1) x = coord(inode,1) sol(idof,1)=sin(2*pi*x) end Essential boundary condition u = 0 at x = 0; bndcon=[1 1 0]
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40 Answers to the exercises of chapter 15 Natural boundary condition: c ∂u/∂x = 0 at x = 1. It is not necessary to prescribe this, because default this condition is set when no essential boundary conditions are prescribed. (b) The same as in item (a) but with different initial conditions. 15.4 (a) For the initial condition add the lines: ii=find(coord>0.0001) % excludes the point at x=0 sol(dest(ii,1))=1000; This change can be made in the file: fem1dcd or in demo_fem1dcd .
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